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Arne Morten Kvarving
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\documentclass[twoside, 11pt, a4paper]{article}
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\usepackage{amsmath,amsfonts,amssymb,graphicx,parskip}
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\usepackage[utf8]{inputenc}
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\DeclareMathOperator{\eps}{\epsilon}
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\newcommand{\dee}{\mathrm{d}}
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\title{A differential equation for approximate wall distance - SplineFEM implementation}
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\author{Arne Morten Kvarving}
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\begin{document}
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\maketitle
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This document describe the derivation of the weak form, and the associated
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Jacobian for the equation given for approximate wall distance in
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\cite{fares}. It is implemented in SplineFEM in the FSWallDistance
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Integrand, see src/Integrands/FSWallDistance.h/C, while
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you can find a test application (and SIM classes) in src/Apps/FSWallDistance.
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The equation is expressed for the inverse wall distance $\mathcal{G}$.
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This field should be initialized to $\mathcal{G}_{\text{init}} \neq 0$ (any
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constant value),
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except on the walls where it should be initialized to
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$\mathcal{G}_{\text{wall}} = \frac{1}{\mathcal{D}_0} = \mathcal{G}_0$.
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The value can be used to simulate rough walls. In any case we must put
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$\mathcal{D}_0 > 0$.
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Initial expression:
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\begin{equation}
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\left(\mathcal{G}\mathcal{G}_x\right)_x + \left(\mathcal{G}\mathcal{G}_y\right)_y+\left(\mathcal{G}\mathcal{G}_z\right)_z + \left(\sigma-1\right)\mathcal{G}\left(\mathcal{G}_{xx}+\mathcal{G}_{yy}+\mathcal{G}_{zz}\right)-\gamma\mathcal{G}^4=0
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\end{equation}
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Using vector notation this is:
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\begin{equation}
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\nabla\cdot\left(\mathcal{G}\nabla \mathcal{G}\right) + \left(\sigma-1\right)\mathcal{G}\nabla^2\mathcal{G} - \gamma\mathcal{G}^4
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\end{equation}
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Multiply with test function, perform integration by parts:
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\begin{eqnarray}
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\int_\Omega \nabla\cdot\left(\mathcal{G}\nabla{G}\right)v\,\dee\Omega + \left(\sigma-1\right)\int_\Omega\mathcal{G}\nabla^2\mathcal{G}v\,\dee\Omega - \gamma\int_\Omega\mathcal{G}^4v\,\dee\Omega = \\
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\int_{\partial\Omega}\mathcal{G}\nabla\mathcal{G}v\cdot \mathbf{n}\,\dee S - \int_\Omega\mathcal{G}\nabla\mathcal{G}\cdot\nabla v\,\dee\Omega +\, \\
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\left(\sigma-1\right)\left(\int_{\partial\Omega}\mathcal{G}\cdot\nabla\mathcal{G}v\cdot\mathbf{n}\,\dee S - \int_\Omega \mathcal{G}\nabla\mathcal{G}\cdot\nabla v\,\dee\Omega-\int_\Omega\nabla^2Gv\,\dee\Omega\right)- \\
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\gamma\int_\Omega\mathcal{G}^4v\,\dee\Omega = 0.
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\end{eqnarray}
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As far as the Jacobian goes, we are looking for
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\begin{equation}
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\frac{\dee}{\dee \mathcal{G}}\left(\nabla \cdot \left(\mathcal{G}\nabla\mathcal{G}\right)+\left(\sigma-1\right)\mathcal{G}\nabla^2\mathcal{G}-\gamma\mathcal{G}^4\right).
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\end{equation}
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We ``cheat'' and insert a delta value;
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\begin{equation}
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\begin{aligned}
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\nabla\cdot\left(\left(\mathcal{G}+\delta \mathcal{G}\right)\nabla\left(\mathcal{G}+\delta \mathcal{G}\right)\right) + \left(\sigma-1\right)\left(\mathcal{G}+\delta \mathcal{G}\right)\nabla^2\left(\mathcal{G}+\delta \mathcal{G}\right)-\gamma\left(\mathcal{G}+\delta \mathcal{G}\right)^4 \Rightarrow \\
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\nabla\cdot\left(\mathcal{G}\nabla \mathcal{G}+\mathcal{G}\nabla\delta\mathcal{G}+\delta\mathcal{G}\nabla\mathcal{G}+\delta\mathcal{G}\nabla\delta\mathcal{G}\right) + \\
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\left(\sigma-1\right)\left(\mathcal{G}\nabla^2\mathcal{G}+\mathcal{G}\nabla^2\delta\mathcal{G}+\delta\mathcal{G}\nabla^2\mathcal{G} + \delta\mathcal{G}\nabla^2\delta\mathcal{G}\right) - \\
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\gamma\left(\mathcal{G}^4+4\mathcal{G}^3\delta\mathcal{G}+6\mathcal{G}^2\delta\mathcal{G}^2 + 4\mathcal{G}\delta\mathcal{G}^3+\delta\mathcal{G}^4\right).
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\end{aligned}
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\end{equation}
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Linearizing, we end up with
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\begin{equation}
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\nabla\cdot\left(\mathcal{G}\nabla\delta\mathcal{G}+\delta\mathcal{G}\nabla\mathcal{G}\right) + \\
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\left(\sigma-1\right)\left(\mathcal{G}\nabla^2\delta\mathcal{G}+\delta\mathcal{G}\nabla^2\mathcal{G}\right) - \\
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\gamma\left(4\mathcal{G}^3\delta\mathcal{G}\right) = 0.
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\end{equation}
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Weak form of the Jacobian, term by term:
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\begin{equation}
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\begin{aligned}
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\int_\Omega \nabla\cdot\left(\mathcal{G}\nabla\delta\mathcal{G}+\delta\mathcal{G}\nabla\mathcal{G}\right)\,\dee\Omega &=&
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\int_{\partial\Omega} \mathcal{G}\nabla\delta\mathcal{G}v\cdot\mathbf{n}\,\dee S - \int_\Omega\mathcal{G}\nabla\delta\mathcal{G}\cdot\nabla v\,\dee\Omega= \\
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&+&\int_{\partial\Omega} \delta\mathcal{G}\nabla\mathcal{G}v\cdot\mathbf{n}\,\dee S - \int_\Omega\delta\mathcal{G}\nabla\mathcal{G}\cdot\nabla v\,\dee\Omega
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\end{aligned}
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\end{equation}
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\begin{equation}
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\begin{aligned}
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\int_\Omega\left(\mathcal{G}\nabla^2\delta\mathcal{G}+\delta\mathcal{G}\nabla^2\mathcal{G}\right)v\,\dee\Omega &=&
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\int_{\partial\Omega}\mathcal{G}\nabla\delta\mathcal{G}v\,\dee S - \int_\Omega\nabla\mathcal{G}\cdot\nabla\left(\delta\mathcal{G}v\right)\,\dee\Omega \\
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&+&\int_{\partial\Omega}\delta\mathcal{G}\nabla\mathcal{G}v\,\dee S - \int_\Omega\nabla\delta\mathcal{G}\cdot\nabla\left(\mathcal{G}v\right)\,\dee\Omega \\
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&=& -\int_\Omega\nabla\mathcal{G}\cdot\nabla\delta\mathcal{G}v\,\dee\Omega - \int_\Omega\delta\mathcal{G}\nabla\mathcal{G}\cdot\nabla v\,\dee\Omega \\
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& & -\int_\Omega\nabla\delta\mathcal{G}\cdot\nabla\mathcal{G}v\,\dee\Omega - \int_\Omega\mathcal{G}\nabla\delta\mathcal{G}\cdot\nabla v\,\dee\Omega
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\end{aligned}
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\end{equation}
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The last term is straight forward;
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\begin{equation}
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\int_\Omega 4\mathcal{G}^3\delta\mathcal{G}v\,\dee\Omega
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\end{equation}
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In summary, our Jacobian reads
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\begin{equation}
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\begin{aligned}
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&-&\int_\Omega\mathcal{G}\nabla\delta\mathcal{G}\cdot\nabla v\,\dee \Omega - \int_\Omega\delta\mathcal{G}\nabla\mathcal{G}\cdot\nabla v\,\dee\Omega \\
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&-&\left(\sigma-1\right)\left(\int_\Omega\nabla\mathcal{G}\cdot\nabla\delta\mathcal{G}v\,\dee\Omega - \int_\Omega\delta\mathcal{G}\nabla\mathcal{G}\cdot\nabla v\,\dee\Omega\right) \\
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&-&\left(\sigma-1\right)\left(\int_\Omega\nabla\delta\mathcal{G}\cdot\nabla\mathcal{G}v\,\dee\Omega - \int_\Omega\mathcal{G}\nabla\delta\mathcal{G}\cdot\nabla v\,\dee\Omega\right) \\
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&-&\gamma\int_\Omega 4\mathcal{G}^3\delta\mathcal{G}v\,\dee\Omega
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\end{aligned}
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\end{equation}
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where
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\[
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\sigma \geq 0.2, \gamma = 1+2\sigma.
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\]
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\bibliographystyle{plain}
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\bibliography{references}
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\end{document}
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@article{Fares,
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title = {A differential equation for approximate wall distance},
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author = {{E. Fares and W. Schröder}},
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journal = {International journal for numerical methods in fluids},
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year = {2002},
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volume = {39},
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pages = {743--762}
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}
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@ -1,191 +0,0 @@
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\documentclass[twoside, 11pt, a4paper]{article}
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\usepackage{amsmath,amsfonts,amssymb,graphicx,parskip}
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\usepackage[utf8]{inputenc}
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\usepackage{subfigure}
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\DeclareMathOperator{\eps}{\epsilon}
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\newcommand{\dee}{\mathrm{d}}
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\title{Spalart-Allmaras turbulence model - IFEM implementation}
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\author{Arne Morten Kvarving}
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\begin{document}
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\maketitle
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This document describe the derivation of the weak form, and the associated
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Jacobian, for the Spalart-Allmaras turbulence model given in \cite{sa}.
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It is implemented in SplineFEM in the SpalartAllmaras
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Integrand, see src/Integrands/SpalartAllmaras.h/C, while
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you can find a test application (and SIM classes) in Apps/SpalartAllmaras.
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\section{Equations}
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Initial expression:
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\begin{equation}
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\frac{\partial\tilde{\nu}}{\partial t} + \mathbf{u}\cdot\nabla\tilde{\nu} = c_{b1}\tilde{S}\tilde{\nu}+\frac{1}{\sigma}\left(\nabla\cdot\left(\nu + \tilde{\nu}\right)\nabla\tilde{\nu} + c_{b2}\left|\nabla\tilde{\nu}\right|^2\right) - c_{w1}f_w\left(\frac{\tilde{\nu}}{d}\right)^2.
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\end{equation}
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Here $\tilde{S}$ models the production and is given by
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\begin{equation}
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\tilde{S} = S + \frac{\tilde{\nu}}{k^2d^2}f_{\tilde{\nu}2}
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\end{equation}
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where $S$ is the normal strain tensor
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\begin{equation}
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\begin{split}
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S &\equiv \sqrt{\hat{\Omega}_{ij}\hat{\Omega}_{ij}}, \\
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\hat{\Omega}_{ij} &\equiv \partial_j u_i - \partial_i u_j, \\
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X &= \frac{\tilde{\nu}}{\nu}, \\
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f_{\tilde{\nu}1} &= \frac{X^3}{X^3+c_{\tilde{\nu}1}^3}, \\
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f_{\tilde{\nu}2} &= 1-\frac{X}{1+Xf_{\tilde{\nu}1}}, \\
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c_{\tilde{\nu}1} &= 7.1, \\
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c_{b1} &= 0.1355, \\
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k &= 0.41.
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\end{split}
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\end{equation}
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The center terms models the diffusion. Here
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\begin{equation}
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\sigma = \frac{2}{3},\ c_{b2} = 0.622
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\end{equation}
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Finally, the last term models destruction. Here
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\begin{equation}
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\begin{split}
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f_w = g\left[\frac{1+c_{w3}^6}{g^6+c_{w3}^3}\right]^{1/6}; \quad g = r + c_{w2}\left(r^6-r\right); \quad r \equiv \frac{\tilde{\nu}}{\tilde{S}k^2d^2} \\
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c_{w1} = c_{b1}/k^2+(1+c_{b2})/\sigma,\quad c_{w2} = 0.3,\quad c_{w3} = 2.
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\end{split}
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\end{equation}
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Furthermore, we have that $\hat{\Omega}_{ij} \equiv \partial_j u_i-\partial_i u_j$.
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That is
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\[
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\hat{\Omega} = \begin{bmatrix}
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0 & \frac{1}{2}\left(\frac{\partial u_1}{\partial x_2}-\frac{\partial u_2}{\partial x_1}\right) & \frac{1}{2}\left(\frac{\partial u_1}{\partial x_3}-\frac{\partial u_3}{\partial x_1}\right) \\
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\frac{1}{2}\left(\frac{\partial u_2}{\partial x_1}-\frac{\partial u_1}{\partial x_2}\right) & 0 & \frac{1}{2}\left(\frac{\partial u_2}{\partial x_3}-\frac{\partial u_3}{\partial x_2}\right) \\
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\frac{1}{2}\left(\frac{\partial u_3}{\partial x_1}-\frac{\partial u_1}{\partial x_3}\right) & \frac{1}{2}\left(\frac{\partial u_3}{\partial x_2}-\frac{\partial u_2}{\partial x_3}\right) & 0
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\end{bmatrix} =
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\begin{bmatrix}
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0 & a & b \\
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-a & 0 & c \\
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-b & -c & 0
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\end{bmatrix}
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\]
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Thus,
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\[
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\hat{\Omega}\hat{\Omega} = 2a^2+2b^2+2c^2
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\]
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and
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\[
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S = 2\sqrt{a^2+b^2+c^2}.
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\]
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\subsection*{Derivation of the Jacobian}
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Temporal term: In this case we consider a BDF1 discretization.
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\subsubsection*{Advection term}
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\bf Strong form:\rm
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\begin{equation}
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\mathbf{u}\cdot\nabla\delta\nu
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\end{equation}
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The weak form is straight forward.
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\subsubsection*{Production term}
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\bf Strong from:\rm
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\begin{equation}
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c_{b1}\hat{S}\delta\nu
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\end{equation}
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The weak form gives a simple mass term.
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\subsubsection*{Diffusion terms}
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\bf Strong form:\rm
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\begin{equation}
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\frac{1}{\sigma}\left(\nabla\cdot\left(\nu+\tilde{\nu}+\delta\nu\right)\nabla\left(\tilde{\nu}+\delta\nu\right)+c_{b2}\left|\nabla\tilde{\nu}+\nabla\delta\nu\right|^2\right)
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\end{equation}
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\bf Linearized, first term:\rm
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\begin{equation}
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\nabla\cdot\nu\nabla\delta\nu + \nabla\cdot\tilde{\nu}\nabla\delta\nu + \nabla\cdot\delta\nu\nabla\tilde{\nu}.
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\end{equation}
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\bf Linearized, second term:\rm
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\begin{equation}
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2\nabla\tilde{\nu}\cdot\nabla\delta\nu
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\end{equation}
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\bf Weak form, first term:\rm
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\begin{equation}
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\begin{aligned}
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\int_\Omega \nabla \cdot \left(\nu\nabla\delta\nu + \tilde{\nu}\nabla\delta\nu + \delta\nu\nabla\tilde{\nu}\right)v\,\dee\Omega &=&
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\int_{\partial\Omega} \left(\nu\nabla\delta\nu + \tilde{\nu}\nabla\delta\nu + \delta\nu\nabla\tilde{\nu}\right)v\cdot \mathbf{n}\,\dee S \\
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&-&\int_\Omega \left(\nu\nabla\delta\nu + \tilde{\nu}\nabla\delta\nu + \delta\nu\nabla\tilde{\nu}\right)\cdot \nabla v\,\dee\Omega.
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\end{aligned}
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\end{equation}
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\bf Weak form, second term:\rm \\
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Straight forward, no integration by parts have to be performed.
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\newpage
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\section{Verification}
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We have performed some verification studies to confirm that the model
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The first one is a two-dimensional backward facing step with a sharp step. The grid and
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geometry definition can be found in Figure \ref{fig:bfsgrid}.
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\begin{figure}[h]
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\begin{center}
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\includegraphics[width=14cm]{bfs2dgrid}
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\end{center}
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\caption{The geometry (and) grid used for BFS2D code validation test. The grid consists of 18751 nodes.}
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\label{fig:bfsgrid}
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\end{figure}
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The boundary conditions for the velocity and pressure are given in Figure \ref{fig:bfsupbc}.
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\begin{figure}[h]
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\begin{center}
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\includegraphics[width=14cm]{bfs2dupbc}
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\end{center}
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\caption{The boundary conditions for the velocity and pressure in the BFS2D code validation test.}
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\label{fig:bfsupbc}
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\end{figure}
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\begin{figure}[h]
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\begin{center}
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\includegraphics[width=8cm]{inflow}
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\end{center}
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\caption{The inflow velocity profile.}
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\label{fig:bfsinflow}
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\end{figure}
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The boundary conditions for the turbulent viscosity are given in Figure \ref{fig:bfsnutbc}.
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\begin{figure}[h]
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\begin{center}
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\includegraphics[width=14cm]{bfs2dnutbc}
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\end{center}
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\caption{The boundary conditions for the turbulent viscosity in the BFS2D code validation test.}
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\label{fig:bfsnutbc}
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\end{figure}
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Some important parameters;
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\[
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\begin{split}
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u_{\infty} &= 1 \\
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\nu &= 0.000331 \\
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\rho &= 1 \\
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Re &= 3025 \\
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\Delta t &= 0.005
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\end{split}
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\]
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The following results were obtained using a second order scheme with SUPG stabilization for both the
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turbulence model and the velocity solver. No substantial differences compared to the results without
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can be observed using the naked eye.
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\newpage
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\begin{figure}[h]
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\begin{center}
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\subfigure[$\frac{x}{S}=4$]{\includegraphics[width=5.5cm]{x0}}
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\subfigure[$\frac{x}{S}=6$]{\includegraphics[width=5.5cm]{x2}}
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\subfigure[$\frac{x}{S}=8$]{\includegraphics[width=5.5cm]{x4}}
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\subfigure[$\frac{x}{S}=10$]{\includegraphics[width=5.5cm]{x6}}
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\subfigure[$\frac{x}{S}=12$]{\includegraphics[width=5.5cm]{x8}}
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\subfigure[$\frac{x}{S}=16$]{\includegraphics[width=5.5cm]{x12}}
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\end{center}
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\caption{Comparison of velocity profiles.}
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\label{fig:bfsx1}
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\end{figure}
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\end{document}
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